Shape Optimization Framework Research Articles

Shape optimization for 2-D mixed-mode fracture using Extended FEM (XFEM) and Level Set Method (LSM)

Weight and service life are often the two most important considerations in design of structural components. This research incorporates a novel crack propagation analysis technique into shape optimization framework to support design of 2-D structural components under mixed-mode fracture for: (1) maximum service life, subject to an upper limit on volume, and (2) minimum weight subject to specified minimum service life. In both cases, structural performance measures are selected as constraints and CAD dimensions are employed as shape design variables. Fracture parameters, such as crack growth rate and crack growth direction are computed using extended finite element method (XFEM) and level set method (LSM). XFEM employs special enrichment functions to incorporate the discontinuity of structural responses caused by the crack surfaces and crack tip fields into finite element approximation. The LSM utilizes level set functions to track the crack during the crack propagation analysis. As a result, this method does not require highly refined mesh around the crack tip nor re-mesh to conform to the geometric shape of the crack when it propagates, which makes the method extremely attractive for crack propagation analysis. An accurate and efficient semi-analytical design sensitivity analysis (DSA) method is developed for calculating gradients of fracture parameters. Two different approaches--a batch-mode, gradient-based, nonlinear algorithm and an interactive what-if analysis--are used for optimization. An engine connecting rod example is used to demonstrate the feasibility of the proposed method.

Convergence of a two-level ideal algorithm for a parametric shape inverse model problem

Similar to the discretization of ordinary or partial differential equations, the numerical approximation of the solution of an optimization problem is possibly subject to numerical stiffness. In the framework of parametric shape optimization, hierarchical representations of the shape can be used for preconditioning, following the idea of Multigrid (MG) methods. In this article, by analogy with the Poisson equation, which is the typical example for linear MG methods, we address a parametric shape inverse problem. We describe the ideal cycle of a two-level algorithm adapted to shape optimization problems that require appropriate transfer operators. With the help of a symbolic calculus software we show that the efficiency of an optimization MG-like strategy is ensured by a small dimension-independent convergence rate. Numerical examples are worked out and corroborate the theoretical results. Applications to antenna design are realized. Finally, some connections with the direct and inverse Broyden–Fletcher–Goldfarb–Shanno preconditioning methods are shown.

Sensitivity of optimal shapes of artificial grafts with respect to flow parameters

The difficulties arising in the numerical solution of PDE-constrained shape optimization problems are manifold. Key ingredients are the optimization strategy and the shape deformation method. Furthermore, the robustness of the optimal shape with respect to simulation parameters is of great interest. In this paper, we consider fluid flows described by the incompressible Navier–Stokes equations. Previous studies on artificial bypass grafts indicated the need for specific constitutive models to account for the non-Newtonian nature of blood; in particular, the constitutive model was shown to affect the solution of the shape optimization problem. We employ a shape optimization framework that couples a finite element solver with quasi-Newton-type optimizers and a Bezier spline shape parametrization. To compute derivatives of the optimal shapes with respect to viscosity, we transform the entire optimization framework by combining the automatic differentiation tools Adifor2 and TAPENADE. We demonstrate the impact of the geometry parametrization and of geometric constraints on the optimization outcome. Finally, we employ the transformed framework to compute the sensitivity of the optimal shape of bypass grafts with respect to kinematic viscosity. The resulting sensitivities predict very accurately the influence of viscosity changes on the optimal shape. The proposed methodology provides a powerful tool to further investigate the necessity of intricate constitutive models by taking derivatives with respect to model parameters.

Development of an efficient aerodynamic shape optimization framework

Although many efforts have been made to develop an aerodynamic shape optimization (ASO) framework, iterative grid generation of the complex configuration within the optimization loop has still been a critical barrier. In this paper, an efficient ASO framework is developed by integrating a parametric grid generator, an optimization toolkit, and a flow solver. A geometry-grid template toolkit is developed to address the need to produce a large number of grids in a timely manner for the parametric design study. An object-oriented optimization toolkit that allows a flexible and extensible interfacing with user-specific codes is used. An in-house full Navier–Stokes flow solver is developed and used in the framework. Code integration is achieved using a black-box interface with script files. Two ASO applications and their optimum solutions are presented to demonstrate the success of this framework.

Isogeometric structural shape optimization

An old dilemma in structural shape optimization is the needed tight link between design model or geometric description and analysis model. The intention of this paper is to show that isogeometric analysis offers a potential and promising way out of this dilemma. To this end we show a structural shape optimization framework based on the isogeometric analysis approach. With the discretization based on Non-Uniform Rational B-Splines (NURBS) the analysis model represents the structural geometry exactly. Furthermore, NURBS enable efficient geometry control together with smooth boundaries. They are the de facto standard in CAD systems, but are also widely used in a shape optimal design context to define the geometry representation and the design variables. With the presented isogeometric approach to shape optimization, the analysis model is inherently merged with the design model, omitting the typically involved interplay between both. We derive analytical sensitivities for NURBS discretizations which allow application of efficient gradient-based optimization algorithms. The present contribution is restricted to two-dimensional problems of linear elasticity, but the extension to three dimensions and other problem classes is straightforward. Some representative examples demonstrate and validate the methodology. Further, the potential of boundary continuity control within isogeometric structural shape optimization is explored to trigger smooth or less smooth (angular) designs.

Soft-Output Demodulation on Frequency-Selective Rayleigh Fading Channels Using AR Channel Models

Optimal link adaption to the scattering function of wide-sense stationary uncorrelated scattering (WSSUS) mobile communication channels is still an unsolved problem despite its importance for next-generation system design. In multicarrier transmission, such link adaption is performed by pulse shaping, i.e., by properly adjusting the transmit and receive filters. For example, pulse-shaped offset-quadrature amplitude modulation (OQAM) systems have recently been shown to have superior performance over standard cyclic prefix orthogonal frequency-division multiplexing (OFDM) (while operating at higher spectral efficiency). In this paper, we establish a general mathematical framework for joint transmitter and receiver pulse shape optimization for so-called Weyl-Heisenberg or Gabor signaling with respect to the scattering function of the WSSUS channel. In our framework, the pulse shape optimization problem is translated to an optimization problem over trace class operators which, in turn, is related to fidelity optimization in quantum information processing. By convexity relaxation, the problem is shown to be equivalent to a convex constraint quasi-convex maximization problem thereby revealing the nonconvex nature of the overall WSSUS pulse design problem. We present several iterative algorithms for optimization providing applicable results even for large-scale problem constellations. We show that with transmitter-side knowledge of the channel statistics a gain of 3-6 dB in signal-to-interference-and-noise-ratio (SINR) can be expected.

The WSSUS Pulse Design Problem in Multicarrier Transmission

Optimal link adaption to the scattering function of wide-sense stationary uncorrelated scattering (WSSUS) mobile communication channels is still an unsolved problem despite its importance for next-generation system design. In multicarrier transmission, such link adaption is performed by pulse shaping, i.e., by properly adjusting the transmit and receive filters. For example, pulse-shaped offset-quadrature amplitude modulation (OQAM) systems have recently been shown to have superior performance over standard cyclic prefix orthogonal frequency-division multiplexing (OFDM) (while operating at higher spectral efficiency). In this paper, we establish a general mathematical framework for joint transmitter and receiver pulse shape optimization for so-called Weyl-Heisenberg or Gabor signaling with respect to the scattering function of the WSSUS channel. In our framework, the pulse shape optimization problem is translated to an optimization problem over trace class operators which, in turn, is related to fidelity optimization in quantum information processing. By convexity relaxation, the problem is shown to be equivalent to a convex constraint quasi-convex maximization problem thereby revealing the nonconvex nature of the overall WSSUS pulse design problem. We present several iterative algorithms for optimization providing applicable results even for large-scale problem constellations. We show that with transmitter-side knowledge of the channel statistics a gain of 3-6 dB in signal-to-interference-and-noise-ratio (SINR) can be expected.